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9 FINITE DIMENSIONAL NORMED SPACES 111
PROBLEMS

1. Let E be the (noncomplete) subspace of the space (c₀) of Banach, consisting of the
sequences x = (£„) of real numbers, having only a. finite number of terms different from

0. For any sequence (a_n) of real numbers, the mapping x -> u(x) = ]T a_n£_n is a linear

n = 0

form on E, and all linear forms on E are obtained in that way; which of them are
continuous (see (5.5.4) and Problem 1 of Section 5.7) ?

2. (a) In a real normed space E, let H be the closed hyperplane of equation u(x) = 0,
where u is a continuous linear form. Show that for any point aeE, the distance

(b) In the space (C_Q) of Banach, let H be the closed hyperplane of equation u(x) =
f; 2-"£, = 0; if a $ H, show that there is no point b e H such that d(a, H) = d(a, b).

3. In a real vector space E, the linear varieties of codimension 1 (Section 5.1 , Problem 5)
are again called hyperplanes ; they are the sets defined by an equation of the type u(x) = a,
where u is a linear form, a any real number; the hyperplanes considered in the text are
those which contain 0, and are also called homogeneous hyperplanes; any hyperplane
defined by an equation u(x) = a is said to be parallel to the homogeneous hyperplane
defined by u(x) — 0. If A is a nonempty subset of E, a hyperplane of support of A is a
hyperplane H defined by an equation u(x) = a, such that u(x) — a 5» 0 for all x e A, or
u(x) — a ^ 0 for all x e A, and u(x₀) — a for at least one point x_Q e A.

(a) In a real normed space E, a hyperplane of support of a set containing an interior
point is closed (see (5.8.2)).

(b) Let A be a compact subset of a real normed space E; show that for any homoge-
neous closed hyperplane H₀ defined by the equation u(x) = 0, there are two hyperplanes
of support of A which are defined by equations of the form u(x) — a, and may eventually
coincide; their distance is at most equal to the diameter of A.

J] 2~"|_n ; show that the closed ball B'(0; 1) has no hyperplane of support having an

nssO

equation of the form u(x) — a (cf. Problem 2(b)).

9. FINITE DIMENSIONAL NORMED SPACES

(5.9.1) Let E be an n-dimensional real (resp. complex) normed vector space;
if(a_l9...₉ #_n) is a basis ofE, the mapping

0/Rⁿ (resp. C¹) onto E is bicontinuous.

We use induction on n, and prove first the result for n = I . We know
by (5.1.5) that ^->^ is continuous; as a_l^Q and \\&i\\ = \\a_t\\ • |£|,
we have |£| <(l/||iil|) •|l&*ill» which proves the continuity of &!-+£,
by (5.5.1).